Chi-Square Critical Value Calculator
Click any cell to highlight the critical value. All values are right-tail critical values χ²α,df.
What Is a Chi-Square Distribution?
Definition
The chi-square (χ²) distribution is a family of right-skewed distributions defined entirely by its degrees of freedom (df). As df increases, the distribution becomes more symmetric and approaches a normal distribution.
Key Properties
χ² values are always ≥ 0. The mean of the distribution equals df, and the variance equals 2×df. It is a special case of the gamma distribution and is always right-tailed in hypothesis testing.
Common Uses
Chi-square tests are used for goodness-of-fit, tests of independence in contingency tables, homogeneity tests, and testing whether a population variance equals a specific value.
How to Read a Chi-Square Table (Step-by-Step)
Reading a chi-square table takes four steps. Once you know your test statistic and degrees of freedom, the table tells you the exact critical value to compare against.
One-Tailed vs Two-Tailed Chi-Square Table
Because the chi-square distribution is always non-negative and right-skewed, the vast majority of chi-square tests are one-tailed (right-tail). Two-tailed tests arise primarily when testing whether a variance equals a specific value.
One-Tailed (Right-Tail) — Most Common
Used in goodness-of-fit, independence, and homogeneity tests. The critical region is entirely in the right tail. Reject H₀ when χ²calc > χ²α,df.
Two-Tailed — Variance Tests
Used when testing H₀: σ² = σ₀². Split α between both tails. Use χ²α/2 for the upper critical value and χ²1−α/2 for the lower critical value.
Chi-Square Table for Contingency Tables
A contingency table chi-square test examines whether two categorical variables are independent. The degrees of freedom are determined by the table dimensions.
Worked Example: 3×2 Contingency Table
You survey 200 people on brand preference across 3 age groups (rows) and 2 preference categories (columns). df = (3−1)×(2−1) = 2. At α = 0.05, the critical value from the chi-square table is 5.991. If your χ² statistic exceeds 5.991, conclude the variables are not independent.
How to Use a Chi-Square Table for Hypothesis Testing
The chi-square table is used in three primary test types. Each follows the same decision rule: reject H₀ if your calculated statistic exceeds the critical value.
Goodness-of-Fit Test
Tests whether observed frequencies match an expected distribution. df = k − 1, where k is the number of categories. Example: testing if a die is fair (6 sides → df = 5).
Test of Independence
Tests whether two categorical variables are related. df = (r−1)(c−1). Example: is there a relationship between gender and product preference?
Variance Test
Tests whether a population variance equals a specified value. df = n − 1. Uses both upper and lower critical values for a two-tailed test.
Chi-Square Table PDF — Free Download
Download a free printable chi-square critical values table in PDF format. All versions include critical values for degrees of freedom 1–100 at standard significance levels.
Chi-Square Critical Values: Common Reference Points
These are the most frequently looked-up chi-square critical values in research and coursework. All values are right-tail critical values at α = 0.05.
Chi-Square Table Statistics: Key Facts & Figures
Frequently Asked Questions About the Chi-Square Table
What is a chi-square table used for?
A chi-square table lists critical values of the χ² distribution for different degrees of freedom and significance levels. It is used to decide whether to reject the null hypothesis in goodness-of-fit tests, tests of independence, and tests of homogeneity.
How do you read a chi-square table?
Find the row matching your degrees of freedom, then move to the column for your significance level (α). The cell at their intersection is the critical value. If your test statistic exceeds this value, reject H₀.
What is the critical value at df=1, p=0.05?
At df=1 and α=0.05, the chi-square critical value is 3.841. This is the most commonly referenced value in statistical testing. If χ²calc > 3.841, reject H₀ at the 5% significance level.
What is the difference between one-tailed and two-tailed chi-square tests?
Most chi-square tests (goodness-of-fit, independence) are one-tailed because the distribution is right-skewed and χ² ≥ 0. Two-tailed tests arise when testing if a variance equals a specific value, where both upper and lower critical values from the table are needed.
How do you calculate degrees of freedom for a chi-square test?
For a goodness-of-fit test: df = (number of categories − 1). For a test of independence: df = (rows − 1) × (columns − 1). For a variance test: df = n − 1.
Can a chi-square value be negative?
No. Chi-square values are always zero or positive because the statistic is calculated by squaring differences: χ² = Σ(O−E)²/E. The distribution ranges from 0 to ∞, which is why only right-tail critical values appear in the table.
What significance level should I use?
α = 0.05 is the standard in most fields. Use α = 0.01 for more rigorous research to reduce Type I error risk. α = 0.10 is acceptable for exploratory or preliminary analyses where missing a real effect (Type II error) is more costly.
What does it mean if my statistic exceeds the critical value?
Reject the null hypothesis. The result is statistically significant — the observed data differs from what you would expect under H₀ by more than can be attributed to chance at your chosen significance level.
Where can I download a chi-square table PDF?
Use the free PDF download links in the section above. Three versions are available: standard (df 1–30), extended (df 1–100), and an annotated reference card with a step-by-step reading guide.
Related Statistical Tables & Resources
Understanding the Chi-Square Table
What Do the α Columns Mean?
Each column header (0.005, 0.01, 0.025, 0.05, 0.10) represents the right-tail area (significance level α). The value in the cell is the chi-square score that cuts off that proportion of the distribution in the right tail.
Relationship Between α and Confidence
α = 0.05 corresponds to a 95% confidence level. A smaller α (e.g., 0.01) means a higher confidence level (99%) and a larger critical value — making it harder to reject H₀ and reducing Type I error.
Why Critical Values Increase with df
As degrees of freedom increase, the chi-square distribution shifts right (mean = df). So the critical value at the same α increases with df — a larger test statistic is needed to achieve significance with more categories.